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Degree of convergence of a function in generalized Zygmund space

  • *Corresponding author: M. Mursaleen

    *Corresponding author: M. Mursaleen 
Abstract Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by
  • In this paper, we obtain the results on the degree of convergence of a function of Fourier series in generalized Zygmund space using deferred Cesàro-generalized Nörlund $ (D^{h}_{g}N^{a,b}) $ transformation. Important corollaries are deduced from our main results. Some applications are also given in support of our main results.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  Degree of convergence of function $ f $

    Figure 2.  Degree of convergence of function $ f $

    Table 1.  Degree of convergence of $f$ for different $n$

    $n$ Degree of convergence of $f$
    100 1.2575829
    1000 1.1507909
    10000 1.1082121
    50000 1.0911295
    100000 1.0854078
    500000 1.0746045
    1000000 1.0707630
    . .
    . .
    $\infty$ 1
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    Table 2.  Degree of convergence of $f$ for different $n$

    $n$ Degree of convergence of $f$
    100 3.8368
    1000 3.5991
    10000 3.4794
    50000 3.4274
    100000 3.4096
    500000 3.3759
    1000000 3.3639
    10000000 3.3315
    100000000 3.3073
    . .
    . .
    $\infty$ 3.1416
     | Show Table
    DownLoad: CSV
  • [1] R. P. Agnew, On deferred Cesàro means, Ann. Math., 33 (1932), 413-421.  doi: 10.2307/1968524.
    [2] C. K. Chui, An Introduction to Wavelets, Wavelet Analysis and its Applications, 1. Academic Press, Inc., Boston, MA, 1992.
    [3] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949.
    [4] S. Lal, Approximation of functions belonging to the generalized Lipschitz class by $C_{1}N_{p}$ summability method of Fourier series, Appl. Math. Comput., 209 (2009), 346-350.  doi: 10.1016/j.amc.2008.12.051.
    [5] S. Lal and A. Mishra, The method of summation $(E, 1)(N, p_{n})$ and trigonometric approximation of function in generalized Hölder metric, J. Indian Math. Soc., 80 (2013), 87-98. 
    [6] B. A. London, Degree of Approximation of Hölder Continuous Functions, Thesis (Ph.D.)-University of Central Florida, 2008.
    [7] Det Kgl. Mollerup, Danske videnskabernes selska, Math.-Fys. Medd., 3 (1920).
    [8] H. K. Nigam, Degree of approximation of a function belonging to weighted $(L_{r}, \xi(t))$ class by $(C, 1)(E, q)$ means, Tamkang J. Math., 42 (2011), 31-37.  doi: 10.5556/j.tkjm.42.2011.514.
    [9] H. K. Nigam and Md. Hadish, Approximation of a function in Hölder class using double Karamata $(K^ {\lambda, \mu})$ method, Eur. J. Pure Appl. Math., 13 (2020), 567-578.  doi: 10.29020/nybg.ejpam.v13i3.3663.
    [10] H. K. Nigam and Md. Hadish, Best approximation of functions in generalized Hölder class, J. Inequal. Appl., (2018), Paper No. 276, 15 pp. doi: 10.1186/s13660-018-1864-y.
    [11] H. K. Nigam and Md. Hadish, Trigonometric approximation of functions by Hausdorff-Matrix product operators, Nonlinear Functional Analysis and Applications, 24 (2019), 675-689. 
    [12] H. K. Nigam and S. Rani, Approximation of function in generalized Hölder class, Eur. J. Pure Appl. Math., 13 (2020), 351-368.  doi: 10.29020/nybg.ejpam.v13i2.3667.
    [13] E. C. TitchmarshThe Theory of Functions, Second edition, Oxford University Press, Oxford, 1939. 
    [14] O. Töeplitz, Uberallagemeine lineara, Mittelbil. Dunger. P.M.F., 22 (2013), 113-119. 
    [15] A. ZygmundTrigonometric Series, 3rd rev. ed., Cambridge University Press, Cambridge, 2002. 
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